Steady state clusters and the Rath-Toth mean field forest fire model

TL;DR

This paper introduces the steady state cluster, a recursive random tree, and constructs a related growth process, conjecturing its connection to the stationary state of the mean field forest fire model, with detailed structural and probabilistic analysis.

Contribution

It defines the steady state cluster and growth process, explores their properties, and links them to the Ráth-Tóth forest fire model through conjectures and structural characterizations.

Findings

The steady state cluster is characterized as a recursive random tree.

The growth process has a fixed-point property and a stationary distribution.

Connections to the Ráth-Tóth forest fire model are conjectured and analyzed.

Abstract

We introduce a random finite rooted tree $\mathcal{C}$, the steady state cluster, characterized by a recursive description: $\mathcal{C}$ is a singleton with probability $1/2$ and otherwise is obtained by joining by an edge the roots of two independent trees $\mathcal{C}'$ and $\mathcal{C}''$, each having the law of $\mathcal{C}$, then re-rooting the resulting tree at a uniform random vertex. We construct a stationary regenerative stochastic process $\mathcal{C}(t)$, the steady state cluster growth process. It is characterized by a simple fixed-point property. Its stationary distribution is the law of the steady state cluster $\mathcal{C}$. We conjecture that $\mathcal{C}(t)$ is the local limit of the evolution of the cluster of a tagged vertex in the stationary state of the mean field forest fire model of R\'ath and T\'oth. We describe its explosions in terms of a L\'evy subordinator, using a state-dependent time change. The steady state cluster is also a multitype Galton-Watson tree with a continuum of types. The steady state cluster conditioned on its size is a random weighted spanning tree of the complete graph equipped random edge weights with a simple explicit joint distribution. The time-reversal of the steady state cluster growth process is realised as the component of a `uniform' vertex in a logging process of a critical multitype Galton-Watson tree conditioned to be infinite. We construct a stationary forest fire model on the infinite rooted tree $\mathbb{Z}^*$ with the property that the evolution of the cluster of the root is a version of the steady state cluster growth process. This model is similar in spirit to Aldous' frozen percolation model on the rooted infinite binary tree. We conjecture that it is the local weak limit of the stationary R\'ath-T\'oth model.